ACTED ON BY NO EXTERNAL FORCES. 767 



competent at any instant to communicate to the rotating body the motion it is then actu- 

 ally endued with, conjoined with the geometrical property of the principal axes that 

 the moment in respect to any one of them of the momenta of the particles of the body 

 due to rotation about either of the other two is zero). 



Consequently from the principle of vis viva, i. e. from the equation 



in addition to the equation 



(cos«)»+(cos/3)*+(cosy)'=l, (1) 



we have the equation 



(coset)* . (cOS|3)'j^(coS7)* M /e)\ 



-^r-+-¥-+-c — 17' ^■'^ 



and the Eulerian system of equations *, 



A^'-(B-CKa,3=0, B^^-(C-A>3^,=0, C^»-(A-BK^,=0, 



* To make this paper complete -witliin itseK so as to come within the comprehension of those who have no pre- 

 vious knowledge of the special problem which it treats, it seems desirable to indicate an elementar)- method of 

 obtaining these ofttimes herein quoted equations. 



1. Suppose no external forces in operation. Consider the effects of the three partial velocities w,, w^, co, in 

 succession as if the others were non-existent. 



Referring to fig. 3, ty, tends to produce no motion about OY or OZ in the tune dt, because the moments of the 

 centrifugal forces about these axes, quantitatively represented by Smzic, "Samy respectively, are each zero by 

 virtue of the geometrical definition of the principal axes. 



Thus to each partial velocity in the time dt is due only a motion of rotation about its own axis. Hence if dy 

 is the variation in y due to to,, 



rfy=ZZ' cos YZl=wAtS^, 

 siny 

 or 



d 008 y= cos ^lUjdt. 



Similarly as regards the variation of cos y due to »j, 



d cos y= — cos aw^dt. 



Hence the total variation d cos y=(co3 ^tu, — cos oM.^dt, 



t. e. 

 or 



l-'-.-(c^-^Y' 



J B-A ,, 

 «»,= — g- WjWjdi, 



with analogous equations for dw.^, diu^. 



When the impressed couples about OX, OY, OZ respectively 'are L, M, "S, tho variations in the angular 

 velocities due to them being 



Idt 'HLdt -Sdt 

 X' -¥-' -C-' 

 these quantities must bo added to the values of dw^, dui^, dm indicated above. Wo have thus tho equations in 

 question. 



It may be as well also here to indicate in the fewest words the rationale of the ellipsoidal reprowntation of 

 the motion. 



A, B, C being the principal moments of inertia, and Aa;'-|-By--|-C;!?=l the equation to the ellipsMd, tho 



6m2 



